Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13191 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C19 | C2×D4 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C2×C22⋊C8 |
| prime radical | C19×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,23)(24,25,28,33,29,32,42,45,30,34,43,48,27,41,36,51)(26,35,38,49,39,47,52,54,40,50,53,55,37,31,44,46);(24,26,36,44,27,37,43,53,30,40,42,52,29,39,28,38)(25,31,45,49,41,50,33,46,34,47,51,55,32,35,48,54);(20,21,22,23)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6179);